JEE PYQ: Motion in a Plane - Question ID 8cc83c23d8e8 (JEE Main 2023)

Core Formula & Concept:

In projectile motion, the horizontal range ($R$) of an object projected with initial velocity $u$ at an angle $\theta$ with the horizontal is given by: $R = \frac{u^2 \sin(2\theta)}{g}$ where:

• $u$ is the initial velocity,
• $\theta$ is the projection angle,
• $g$ is the acceleration due to gravity.

This formula arises from the fact that the horizontal range depends on the horizontal component of velocity ($u \cos \theta$) and the total time of flight ($T = \frac{2u \sin \theta}{g}$). The product of these two quantities yields the range formula above.

The question involves two objects projected with the same speed $u$ but at angles $\alpha$ and $\beta$, where $\alpha + \beta = 90^\circ$. We are to find the ratio of their horizontal ranges.

Step-by-Step Derivation:

Step 1: Write the range expressions for both objects.
For the first object (angle $\alpha$): $R_1 = \frac{u^2 \sin(2\alpha)}{g}$ For the second object (angle $\beta$): $R_2 = \frac{u^2 \sin(2\beta)}{g}$

Step 2: Use the given condition $\alpha + \beta = 90^\circ$.
Since $\alpha + \beta = 90^\circ$, we can express $\beta$ as: $\beta = 90^\circ - \alpha$

Step 3: Substitute $\beta$ into $R_2$.
Now, compute $\sin(2\beta)$: $\sin(2\beta) = \sin(2(90^\circ - \alpha)) = \sin(180^\circ - 2\alpha) = \sin(2\alpha)$ This follows from the trigonometric identity: $\sin(180^\circ - x) = \sin x$.

Step 4: Compare $R_1$ and $R_2$.
Substituting back: $R_2 = \frac{u^2 \sin(2\alpha)}{g} = R_1$ Thus, the ranges are equal: $R_1 : R_2 = 1 : 1$

Common Traps & Exam Tip:

Trap 1: Misapplying the range formula.
Some students mistakenly use $R = \frac{u^2 \sin \theta}{g}$ instead of $R = \frac{u^2 \sin(2\theta)}{g}$. This leads to incorrect results. Always double-check the formula for range. Trap 2: Overlooking the trigonometric identity.
Students may fail to recognize that $\sin(2\beta) = \sin(2\alpha)$ when $\alpha + \beta = 90^\circ$. This step is crucial for simplifying the ratio. Remember that $\sin(180^\circ - x) = \sin x$. Trap 3: Assuming unequal ranges due to different angles.
Intuitively, one might think that different angles lead to different ranges. However, the condition $\alpha + \beta = 90^\circ$ ensures that the ranges are equal, as shown in the derivation. Always rely on the math rather than intuition alone. Exam Tip:
When dealing with projectile motion problems involving complementary angles (i.e., $\alpha + \beta = 90^\circ$), remember that the ranges will be equal. This is a common pattern in JEE problems and can save time during the exam.